Combinatorics Calculator
How many arrangements or selections are possible? Permutations and combinations with and without repetition from n and k, each with its formula.
Arrangements without repetition
Result
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Formula
P(n,k) = n! / (n-k)!
Note: These calculations are for informational purposes only and do not replace professional tax or financial advice. All information without guarantee.
FAQ
Frequently Asked Questions
What is the difference between permutations and combinations?
Permutations count arrangements where order matters (ABC is different from BAC). Combinations count selections where order does not matter (ABC and BAC are the same).
How do I calculate the number of possible combinations?
Combinations: C(n,k) = n! / (k! times (n-k)!). For example, choosing 3 items from 10: C(10,3) = 120. The exclamation mark denotes factorial (e.g., 5! = 120).
Which of the four options do I choose – with or without repetition?
It depends on whether order matters and whether an element may be chosen more than once. Order matters: a podium of 3 out of 10 runners is calculated with “Permutation P(n,k)” (720), a four-digit PIN from 10 digits with “Permutation with rep.” (10^4 = 10,000). Order doesn't matter: a 6-out-of-49 lottery draw is a “Combination C(n,k)” (13,983,816), and 3 scoops of ice cream from 5 flavors, where the same flavor may be chosen more than once, is a “Combination with rep.” (35).
Guide
What is the Combinatorics Calculator?
The combinatorics calculator computes permutations, combinations and variations with and without repetition.
How does the Combinatorics Calculator work?
Choose the calculation type: permutation (arrangement of all elements), combination (selection without order) or variation (selection with order). The calculator computes the count and shows the formula.
Key Data and Facts
Permutation: n! Combination without repetition: n! / (k! * (n-k)!). Variation without repetition: n! / (n-k)!. Factorial: n! = 1 * 2 * ... * n.
Step-by-Step Guide
How to use the combinatorics calculator step by step: 1. Choose the type of calculation – permutation (arrange all elements), combination (selection without order) or variation (selection with order). 2. Enter the parameters – n = total number of elements, k = number of elements to be selected. 3. Set repetition – with repetition means that elements may be chosen multiple times. 4. Understand the formulas: permutation without repetition = n!. Combination without repetition = n! / (k! * (n-k)!), also known as the binomial coefficient (n choose k). Variation without repetition = n! / (n-k)!. 5. Practical applications: calculate lottery probabilities (6 out of 49 = 13,983,816 combinations), plan seating arrangements, count PIN codes (4 digits with repetition = 10,000 possibilities). Tip: the factorial grows extremely fast – 10! = 3,628,800, 20! already has 19 digits.
Calculation Example
Combination of 6 from 49 (lottery): 49! / (6! * 43!) = 13,983,816. Variation of 3 from 10 without repetition: 10! / 7! = 720.
Sources
Official sources
Calculations are based on applicable German laws and official data:
Full methodology at Methodology.
Reviewed by Konstantin Iakovlev · Last updated:
Mathematics
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